16.333
Prof. J. P. How
Handout #4
Oct 26, 2004
Due: Nov 16, 2004
16.333 Homework Assignment #4
Please include all code used to solve these problems.
1. Given the following model of the aerosonde vehicle extracted from the (trim condition,
level flight at 23 m/s at sea level)
Guδe (s) =
Gαδe (s) =
Gqδe (s) =
Gθδe (s) =
Guδta (s) =
Gαδta (s) =
Gqδta (s) =
Gθδta (s) =
152.2s + 913.1
Δ(s)
−25.72s2 − 2.439s − 3.85
Δ(s)
3
−24.18s − 99.13s2 − 9.672s
Δ(s)
2
−24.18s − 99.13s − 9.672
Δ(s)
1.236s3 + 10.46s2 + 131.4s + 38.5
Δ(s)
2
−1.192s + 0.3228s − 0.1366
Δ(s)
3
−0.7646s + 3.058s
Δ(s)
−0.7646s2 + 3.058
Δ(s)
where
Δ(s) = s4 + 8.28s3 + 105.1s2 + 14.22s + 24.29
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to capture the engine lag. The elevators on this
and δta = H(s)δtc , with H(s) = s+3
aircraft are fast enough that they can be ignored.
Part of the system model available on­line is shown in the figure. u and α are available
as the first and third outputs of VelW, q is available as the 5th element of States, and
θ is available as the second element of Euler. The elevator is the second control input,
the throttle (δtc ) is the fourth. Note that a very simple roll loop has been added to
facilitate analysis of the longitudinal dynamics.
• Compare the OL dynamics to the 747 ­ any surprises?
• Given these dynamics, design an altitude autopilot
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• Use classical, multi­loop closure techniques as discussed in class
• Use Full state feedback. Discuss you rationale for where to locate the regulator
poles.
• Use output feedback with u, θ, and α measurements. Discuss you rationale for
where to locate the estimator poles.
• Use the simplified block diagram available on the class web page, implement the
controller in simulink. Compare the nonlinear simulation response with your
predictions ­ any surprises?
2. Continue Question #4 of HW3, but this time use the short period model and design
the controller using state space techniques. As part of this design,
(a) Develop a full state feedback controller that puts the regulator poles where re­
quired.
(b) Then develop a closed­loop estimator for the system assuming that you can mea­
sure the pitch angle θ. Choose estimator pole locations that have the same imag­
inary part as the regulator poles, but a real part that is 3–4 times larger (in
magnitude).
(c) Put the regulator and estimator together to form the compensator Gc (s) that
maps y = θ to u = δe (recall that actually u = −Gc y). To implement this design,
perform the same trick that we did in the notes, and use u = Gc e, where e = θc −θ.
You should now be able to perform closed­loop simulations of the response of the
system to a step in θc .
(d) Check your pole locations on the full set of longitudinal dynamics. Is the response
stable?
(e) Compare the frequency response of this state space controller and the controller
that you designed in HW3.
3. Using the same approach given in class, design a heading autopilot for the F­4C (i.e.
one that can track a given Ψd ) using the dynamics in condition 2. Assume that the
actuator servo dynamics has the transfer function Hs (s) = 20/(s + 20).
(a) This design will consist of a yaw damper, a roll controller, and a feedback on the
heading ψ.
(b) Simulate the response to an interesting Ψ sequence (like a sequence of 45 deg
turns) and comment on the performance. Include a limiter on the desired bank
angle of ±15 degs.
4. Write a brief summary of the paper by Vincenti that was handed out. In particular,
be sure state his main point and whether or not you agree with it.
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Figure 1: Simplified block diagram for the Aerosonde
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